Lecture 35: Chapter 13, Section 2
Two Quantitative Variables
Interval Estimates
PI
for Individual Response, CI for Mean Response
Explanatory Value Close to or Far from Mean
Approximating Intervals by Hand
Width of PI vs. CI
Guidelines for Regression Inference
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Learning
Elementary Statistics: Looking at the Big Picture
1
Looking Back: Review

4 Stages of Statistics




Data Production (discussed in Lectures 1-4)
Displaying and Summarizing (Lectures 5-12)
Probability (discussed in Lectures 13-20)
Statistical Inference
 1 categorical (discussed in Lectures 21-23)
 1 quantitative (discussed in Lectures 24-27)
 cat and quan: paired, 2-sample, several-sample
(Lectures 28-31)
 2 categorical (discussed in Lectures 32-33)
 2 quantitative
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Elementary Statistics: Looking at the Big Picture
L35.2
Correlation and Regression (Review)

Relationship between 2 quantitative variables


Display with scatterplot
Summarize:
Form: linear or curved
 Direction: positive or negative
 Strength: strong, moderate, weak
If form is linear, correlation r tells direction and strength.
Also, equation of least squares regression line lets us
predict a response for any explanatory value x.

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Elementary Statistics: Looking at the Big Picture
L35.3
Population Model; Parameters and Estimates
Summarize linear relationship between sampled x and y
values with line
minimizing sum of
squared residuals
. Typical residual size is
Model for population relationship is
and responses vary normally with standard deviation
 Use
to estimate
Looking Back: Our hypothesis
 Use
to estimate
test focused on slope.
 Use
to estimate
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Elementary Statistics: Looking at the Big Picture
L35.4
Regression Null Hypothesis (Review)

no population relationship between x and y
Test statistic t
P-value is probability of t this extreme, if true
(where t has n-2 df)
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Elementary Statistics: Looking at the Big Picture
L35.5
Confidence Interval for Slope (Review)
Confidence interval for
is
where multiplier is from t dist. with n-2 df.
If n is large, 95% confidence interval is
If CI does not contain 0, reject
x and y are related.
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Elementary Statistics: Looking at the Big Picture
, conclude
L35.6
Interval Estimates in Regression
Seek Prediction and Confidence Intervals for
 Individual response to given x value (PI)


For large n, approx. 95% PI:
Mean response to subpopulation with given
x value (CI)
For large n, approx. 95% CI:
Both intervals centered at predicted y-value
These approximations may be poor if n is small or
if given x value is far from average x value.

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Elementary Statistics: Looking at the Big Picture
L35.7
Example: Reviewing Data in Scatterplot



Background: Property owner feels reassessed value $40,000
of his 4,000 sq.ft. lot is too high. For random sample of 29
local lots, means are 5,619 sq.ft. for size, $34,624 for value.
Regression equation y^ =1,551+5.885x, r =+0.927, s=$6,682.
Question: Where would his property appear on scatterplot?
Response: shown in red (noticeably high for x = 4,000)
*
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A Closer Look: His lot is smaller
than average but valued higher
than average; some cause for
concern because the relationship
is strong and positive. But it’s not
perfect, so we seek statistical
evidence of an unusually high
value for the lot’s size.
Elementary Statistics: Looking at the Big Picture
L35.8
Example: An Interval Estimate



Background: Property owner feels reassessed value $40,000
of his 4,000 sq.ft. lot is too high. For random sample of 29
local lots, means are 5,619 sq.ft. for size, $34,624 for value.
Regression equation y^ =1,551+5.885x, r =+0.927, s=$6,682.
Questions: What range of values are within two standard
errors of the predicted value for 4,000 sq.ft.? Does $40,000
seem too high?
Responses: Predict ^y =$1,551+$5.885(4,000) = $25,091.
Approximate range of plausible values for individual 4,000
sq.ft. lot is $25,091  2($6,682) = ($11,727, $38,455).
Yes: $40,000 is completely above the interval.
©2011 Brooks/Cole,
Cengage Learning
Elementary Statistics: Looking at the Big Picture
Practice: 13.18c p.658
L35.10
Example: Interval Estimate on Scatterplot



Background: A homeowner’s 4,000 sq.ft. lot is assessed at
$40,000. Predicted value is $25,091 and predicted range of
values is ($11,727, $38,455).
Question: Where do the prediction and range of values
appear on the scatterplot?
Response: prediction shown in orange, range in blue
*
*
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Elementary Statistics: Looking at the Big Picture
L35.12
Prediction Interval vs. Confidence Interval

Prediction interval corresponds to 68-95-99.7 Rule
for data: where an individual is likely to be.


Confidence interval is inference about mean: range
of plausible values for mean of sub-population.


PI is wider: individuals vary a great deal
CI is narrower: can estimate mean with more precision
Both PI and CI in regression utilize info about x to
be more precise about y (PI) or mean y (CI).
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Elementary Statistics: Looking at the Big Picture
L35.14
Examples: Series of Estimation Problems

Based on sample of male weights, estimate

weight of individual male
No regression
needed.

mean weight of all males
Based on sample of male hts and weights, estimate

weight of individual male, 71 inches tall

mean weight of all 71-inch-tall males

weight of individual male, 76 inches tall

mean weight of all 76-inch-tall males
Examples use data from sample of college males.
}

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Elementary Statistics: Looking at the Big Picture
L35.17
Example: Estimate Individual Wt, No Ht Info



Background: A sample of male weights have
mean 170.8, standard deviation 33.1. Shape of
distribution is close to normal.
Question: What interval should contain the weight
of an individual male?
Response: Need to know distribution of weights is
approximately normal to apply 68-95-99.7 Rule:
Approx. 95% of individual male weights in interval
170.8  2(33.1) = (104.6, 237.0)
©2011 Brooks/Cole,
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Elementary Statistics: Looking at the Big Picture
Practice: 13.20e p.659
L35.18
Examples: Series of Estimation Problems


Based on sample of male weights, estimate

weight of individual male

mean weight of all males
Based on sample of male heights and weights, est

weight of individual male, 71 inches tall

mean weight of all 71-inch-tall males

weight of individual male, 76 inches tall

mean weight of all 76-inch-tall males
©2011 Brooks/Cole,
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Elementary Statistics: Looking at the Big Picture
L35.20
Example: Estimate Mean Wt, No Ht Info



Background: A sample of 162 male weights have mean
170.8, standard deviation 33.1.
Questions:

What interval should contain the mean weight of all
males?

How does it compare to this interval for an individual
male’s weight?
Responses:

Need to know sample size n to construct approximate
95% confidence interval for mean:

Interval for mean involves division by square root of n
narrower than interval for individual
©2011 Brooks/Cole,
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Elementary Statistics: Looking at the Big Picture
Practice: 13.20f p.659
L35.22
Examples: Series of Estimation Problems


Based on sample of male weights, estimate

weight of individual male

mean weight of all males
Based on sample of male heights and weights, est

weight of individual male, 71 inches tall
Need

mean weight of all 71-inch-tall males
regression

weight of individual male, 76 inches tall

mean weight of all 76-inch-tall males
©2011 Brooks/Cole,
Cengage Learning
Elementary Statistics: Looking at the Big Picture
L35.24
Examples: Series of Estimation Problems
The next 4 examples make use of regression on
height to produce interval estimates for weight.
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Elementary Statistics: Looking at the Big Picture
L35.26
Example: Predict Individual Wt, Given Av. Ht



Background: Male hts: mean about 71 in. Wts: s.d. 33.1 lbs.
Regression of wt on ht has r =+0.45, p=0.000. Regression
line is
. and s = 29.6 lbs.
Questions: How much heavier is a sampled male, for each
additional inch in height? Why is
? What interval
should contain the weight of an individual 71-inch-tall male?
(Got interval estimates for x=71.)
Responses:

For each additional inch, sampled male weighs 5 lbs more.

.
because wts vary less about line than about mean.

Look at PI for x = 71: (114.20, 231.47).
©2011 Brooks/Cole,
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Elementary Statistics: Looking at the Big Picture
Practice: 13.41a-b p.667 L35.27
Example: Approx. Individual Wt, Given Av. Ht



Background: Male hts: mean about 71 in. Wts: s.d. 33.1 lbs.
Regression of wt on ht has r =+0.45, p=0.000. Regression
line is
. and s = 29.6 lbs. Got interval
estimates for wt when ht=71:
Questions:

How do we approximate interval estimate for wt. of an
individual 71-inch-tall male by hand?

Is our approximate close to the true interval?
Responses:

Predict y for x=71:

Close? Yes.
©2011 Brooks/Cole,
Cengage Learning
Elementary Statistics: Looking at the Big Picture
Practice: 13.18c-d p.658
L35.29
Examples: Series of Estimation Problems


Based on sample of male weights, estimate

weight of individual male

mean weight of all males
Based on sample of male heights and weights, est

weight of individual male, 71 inches tall

mean weight of all 71-inch-tall males

weight of individual male, 76 inches tall

mean weight of all 76-inch-tall males
©2011 Brooks/Cole,
Cengage Learning
Elementary Statistics: Looking at the Big Picture
L35.31
Example: Est Mean Wt, Given Average Ht


Background: Male hts: mean about 71 in. Wts: s.d. 33.1 lbs.
Regression of wt on ht has r =+0.45, p=0.000. Regression
line is
. and s = 29.6 lbs.
Questions:

What interval should contain mean weight of all 71-inchtall males?
How do we approximate the interval by hand? Is it close?
Response:

SoftwareCI for x=71

Predict y for x=71:


Close? Yes!
©2011 Brooks/Cole,
Cengage Learning
Elementary Statistics: Looking at the Big Picture
Practice: 13.18e-f p.658
L35.33
Examples: PI and CI for Wt; Ht=71 Inches
PI
CI
*
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Elementary Statistics: Looking at the Big Picture
L35.35
Example: Estimate Wt, Given Tall vs. Av. Ht

Background: Regression of male wt on ht
produced equation
.
For height 71 inches, estimated weight is

Question: How much heavier will our estimate be
for height 76 inches?
Response: Since slope is about 5, predict 5 more
lbs for each additional inch; 25 more lbs for 76,
which is 5 additional inches:
Instead of weight about 173, estimate weight about
173+25=198.

©2011 Brooks/Cole,
Cengage Learning
Elementary Statistics: Looking at the Big Picture
Practice: 13.21b p.659
L35.36
Examples: Series of Estimation Problems


Based on sample of male weights, estimate

weight of individual male

mean weight of all males
Based on sample of male heights and weights, est

weight of individual male, 71 inches tall

mean weight of all 71-inch-tall males

weight of individual male, 76 inches tall

mean weight of all 76-inch-tall males
©2011 Brooks/Cole,
Cengage Learning
Elementary Statistics: Looking at the Big Picture
L35.38
Example:Find Individual Wt, Given Tall Ht

Background: Regression of male weight on height has r
=+0.45, p=0.000strong evidence of moderate positive
relationship. Reg. line
and s=29.6 lbs. Got interval estimates for x=76.

Questions: What interval should contain the weight of an
individual male, 76 inches tall? How does the interval
compare to the one for ht=71?

Responses:

PI for x=76

Predicted wt (fit) about 25 lbs more for x=76 than for 71:
198.21-172.83=25.38 (5 more lbs per additional inch).
©2011 Brooks/Cole,
Cengage Learning
Elementary Statistics: Looking at the Big Picture
Practice: 13.20c p.659
L35.40
Example: Approx. Individual Wt for Tall Ht

Background: Regression of male weight on height
has r =+0.45, p=0.000strong evidence of moderate
positive relationship. Reg. line
and s=29.6 lbs. Got interval estimates for x=76.

Questions:



How do we approximate the prediction interval by hand?
Is it close to the true interval?
Responses:

Predict y for x=76:

Close? Yes!
©2011 Brooks/Cole,
Cengage Learning
Elementary Statistics: Looking at the Big Picture
L35.42
Examples: Series of Estimation Problems


Based on sample of male weights, estimate

weight of individual male

mean weight of all males
Based on sample of male heights and weights, est

weight of individual male, 71 inches tall

mean weight of all 71-inch-tall males

weight of individual male, 76 inches tall

mean weight of all 76-inch-tall males
©2011 Brooks/Cole,
Cengage Learning
Elementary Statistics: Looking at the Big Picture
L35.44
Example: Est Mean Wt, Given Tall Ht



Background: Regression of 162 male wts on hts has r
=+0.45, p=0.000strong evidence of moderate positive
relationship. Reg. line
and s=29.6 lbs. Got interval estimates for x=76.
Questions:

What interval should contain mean wt of all 76-in males?

How do we approximate the interval by hand? Is it close?
Responses:

Refer to CI.

Predict y for x=76:
Close? Not very.
©2011 Brooks/Cole,
Cengage Learning
Elementary Statistics: Looking at the Big Picture
Practice: 13.21d-e p.659
L35.46
Examples: PI and CI for Wt; Ht=71 or 76
PI
PI
CI
*
CI
*
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Elementary Statistics: Looking at the Big Picture
L35.48
Interval Estimates in Regression (Review)
Seek interval estimates for
 Individual response to given x value (PI)


Mean response to subpopulation with
given x value (CI)


For large n, approx. 95% PI:
For large n, approx. 95% CI:
Intervals approximately correct only for x
values close to mean; otherwise wider

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Especially CI much wider for x far from mean
Elementary Statistics: Looking at the Big Picture
L35.49
PI and CI for x Close to or Far From Mean
:
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Elementary Statistics: Looking at the Big Picture
L35.50
Examples: Solutions to Estimation Problems
wt of individual male (104.6, 237.0)

mean wt of all males (165.6, 176.0)
CI much narrower for mean: width 10.4 instead of 132.4

wt of individual, ht=71 (114.2, 231.5)
PI narrower given ht info: width 117.3 instead of 132.4

mean wt, ht=71
(168.2, 177.5)
CI much narrower for mean, also since ht given: width 9.3

wt of individual, ht=76 (139.0, 257.5)
PI wider than for x=71 since 76 far from mean: 118.5 not 117.3

mean wt, ht=76
(188.6, 207.8)
CI much wider than for x=71 since 76 far from mean:19.2 not 9.3

©2011 Brooks/Cole,
Cengage Learning
Elementary Statistics: Looking at the Big Picture
L35.51
Summary of Example Intervals
CI always
narrower
than PI
:
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Elementary Statistics: Looking at the Big Picture
L35.52
Summary of Example Intervals
:
CI and PI
can be
narrower if
x info given
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Elementary Statistics: Looking at the Big Picture
L35.53
Summary of Example Intervals
:
CI and PI
centered at
heavier wt
for taller ht
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Elementary Statistics: Looking at the Big Picture
L35.54
Summary of Example Intervals
:
CI and PI
wider for wt
far from
average
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Elementary Statistics: Looking at the Big Picture
L35.55
Example: A Prediction Interval Application



Background: A news report stated that Michael Jackson was
a fairly healthy 50-year-old before he died of an overdose.
“His 136 pounds were in the acceptable range for a 5-foot-9
man…”
Question: Based on the regression equation equation
and
and s=29.6 lbs, would we agree that
136 lbs. is not an unusually low weight?
Response: For x = 69, predict y = -188+5.08(69)=162.52.
Our PI is 162.52  2(29.6) = (103.32, 221.72); his weight 136
certainly falls in the interval.
A Closer Look: Our PI is a bit misleading because the distribution of weights is
actually somewhat right-skewed, not normal. More of the spread reported in
s=29.6 comes about from unusually heavy men, and less from unusually light men.
©2011 Brooks/Cole,
Cengage Learning
Elementary Statistics: Looking at the Big Picture
L35.56
Example: A Prediction Interval Application
A Closer Look: Our PI is a bit misleading because the distribution of weights is
actually somewhat right-skewed, not normal. More of the spread reported in
s=29.6 comes about from unusually heavy men, and less from unusually light men.
©2011 Brooks/Cole,
Cengage Learning
Elementary Statistics: Looking at the Big Picture
L35.58
Guidelines for Regression Inference







Relationship must be linear
Need random sample of independent observations
Sample size must be large enough to offset nonnormality
Need population at least 10 times sample size
Constant spread about regression line
Outliers/influential observations may impact
results
Confounding variables should be separated out
©2011 Brooks/Cole,
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Elementary Statistics: Looking at the Big Picture
L35.59
Lecture Summary
(Inference for QuanQuan; PI and CI)

Interval estimates in regression: PI or CI





Non-regression PI (individual) and CI (mean)
Regression PI and CI for x value near mean or far
Approximating intervals by hand
Width of PI vs. CI
Guidelines for regression inference
©2011 Brooks/Cole,
Cengage Learning
Elementary Statistics: Looking at the Big Picture
L35.60